Hidden extreme multistability in memristive hyperchaotic system

Abstract

By utilizing a memristor to substitute a coupling resistor in the realization circuit of a three-dimensional chaotic system having one saddle and two stable node-foci, a novel memristive hyperchaotic system with coexisting infinitely many hidden attractors is presented. The memristive system does not display any equilibrium, but can exhibit hyperchaotic, chaotic, and periodic dynamics as well as transient hyperchaos. In particular, the phenomenon of extreme multistability with hidden oscillation is revealed and the coexistence of infinitely many hidden attractors is observed. The results illustrate that the long term dynamical behavior closely depends on the memristor initial condition thus leading to the emergence of hidden extreme multistability in the memristive hyperchaotic system. Additionally, hardware experiments and PSIM circuit simulations are performed to verify numerical simulations.

Introduction

Conventionally, a nonlinear dynamical system described by a set of autonomous ordinary differential equations can be easily implemented with an electronic circuit via standard op-amp integrators and analogue multipliers, where an op-amp integrator linked with a feedback capacitor and some inverting input resistors performs the mathematical operation of integration with respect to time, and a functionally complete four-quadrant analogue multiplier achieves the mathematical operation of nonlinearity [1], [2]. Memristor [3], a fundamental two-terminal electronic element with an adjustable resistance or conductance [4], has potential applications in various nonlinear dynamical circuits due to the special features of both nonlinearity and memory. In recent years, by introducing memristors into existing oscillating circuits or substituting nonlinear resistors in classical chaotic circuits with memristors, a variety of memristor based chaotic/hyperchaotic circuits are simply established and broadly investigated [1], [5], [6], [7]. In this paper, based on the realization circuit of a three-dimensional chaotic system, a systematic circuit realization scheme of a four-dimensional memristor based hyperchaotic system is proposed by utilizing a memristor to substitute a linear coupling resistor.

The conventional self-excited attractors, such as Lorenz attractor [8], Chen attractor [9], Lü attractor [10], and many other widely-known attractors [11], [12], [13], [14], are all excited from unstable index-2 saddle-foci, namely, an attractor with an attraction basin corresponds to an unstable equilibrium. Consequently, self-excited attractors can easily be implemented by disposing unstable index-2 saddle-foci in terms of added breakpoints in the model system [15], [16]. However, a new type of attractors, defined as hidden attractors [17], have been first found in classical Chua's circuit [18], [19], whose basin of attraction does not intersect with small neighborhoods of the equilibria of the system [17]. Due to the existences of hidden attractors, some particular dynamical systems associated with line equilibrium, or no equilibrium, or stable equilibrium have attracted much attention recently [20], [21], [22], [23], [24], [25], [26]. In particular, to the extent that they have been known to exist, dynamical systems with no equilibrium have mostly been considered as unphysical or mathematically incomplete. However, as experience shows, a system that presents hidden dynamical behavior doesn't need to also display an unstable equilibrium state [23], [24], [25], [26]. In this paper, a memristor based hyperchaotic system with no equilibrium is proposed, from which the complex and striking phenomenon of coexisting infinitely many hidden attractors’ behavior and the corresponding hidden extreme multistability are perfectly revealed. To the best knowledge of the authors, this phenomenon of coexisting infinitely many hidden attractors has not been reported in any literatures.

Multistability, meaning the coexistence of many different kinds of attractors, is an intrinsic property of many nonlinear dynamical systems and has become very important research topic and received much attention recently [27], [28], [29], [30], [31], [32]. Multistability exhibits a rich diversity of stable states of a nonlinear dynamical system and makes the system offer a great flexibility. Particularly, when the number of coexisting attractors generating from a dynamical system tends to infinite, the coexistence of infinitely many attractors depending on the initial condition of a certain state variable is alleged extreme multistability [33], which has been reported in two unidirectionally coupled Lorenz systems [34], two bi-directionally coupled Rössler oscillator with partial synchronization [33], [35], and several memristive circuit with ideal active flux-controlled memristors [36], [37], [38]. Since multistability can be used for image processing [39] or regarded as an additional source of randomness using for many information engineering applications [32], it is attractive to seek for a memristor based hyperchaotic system that has the striking dynamical behavior of infinitely many attractors. Actually, in a memristor based chaotic circuit [40], [41] or a chaotic memory system [2], the complex dynamical behaviors are dependent on the initial states of the memristor or the memory element, which just reflect the emergences of extreme multistability in these memory systems.

In this paper, by utilizing a newly proposed circuit realization scheme, a novel memristive hyperchaotic system is derived from an existing three-dimensional chaotic system [42], from which the coexisting phenomenon of infinitely many hidden attractors, i.e., hidden extreme multistability, is discovered and then revealed. It is important to note, that different from extreme multistability emerged from line equilibrium reported in Refs. [36], [37], [38], the infinitely many hidden attractors found in this paper are not related to any equilibrium. The paper is organized as follows. In Section 2, a circuit realization scheme is proposed and a memristive hyperchaotic system with no equilibrium is then constructed. In Section 3, with system parameters varying, bifurcation diagrams and the corresponding Lyapunov exponent spectra are plotted to reveal hidden hyperchaotic dynamical behaviors. Additionally, transient hyperchaotic behavior is also illustrated. In Section 4, with memristor initial condition varying, bifurcation diagram, Lyapunov exponent spectra, and phase portraits are given, upon which infinitely many hidden attractors’ dynamics and transient transition behavior are demonstrated. Experimental measurements and PSIM simulations are performed in Section 5 to validate numerical simulations. The conclusions are summarized in Section 6.